Suppose $R$ is a local ring and let $I\subset R$ be some nontrivial ideal. Are there conditions that we can place on $I$ so that if $R/I$ is regular, then so is $R$?
I am aware of the result that states: if $R$ is already a regular local ring, then $R/I$ is regular iff $I$ is generated by a subset of a regular system of parameters. But I am wondering more about whether or not the regularity of $R$ itself can be determined by it's quotient $R/I$.
Background for this question: after reading the responses to the question When is a blow-up non-singular? I am trying to work through the first argument in the second section of the paper "On the smoothness of blow-ups." I think the author uses the result that I am asking about when they state: "$S_P/f_i S_P$ is a regular local ring; since $f_i$ is a non-zero-divisor on $S_P$, it follows that $S_P$ is itself regular." I am just trying to figure out where this result comes from.